How to color an object when touched by animated character?
This sounds right down the alley of dynamic paint.
To get a tile to light up, I recommend using an image canvas with single pixels for each tile, and when it goes into the material, set the interpolation to nearest neighbor (no interpolation). As for the wave effect, that should be fairly easy as well with a second canvas set to wave. In the floor shader, you can use the light from the first canvas to control the appearance of the waves from the second so they don’t go outside the lit tile.
I have not tested this.
For more information, check out dynamic paint in the physics section of the manual at docs.blender.org